Recent work has highlighted remarkable effects of classical thermal fluctuations in the dipolar spin ice compounds, such as “artificial magnetostatics,” manifesting as Coulombic power-law spin correlations and particles behaving as diffusive “magnetic monopoles.” In this paper, we address quantum spin ice, giving a unifying framework for the study of magnetism of a large class of magnetic compounds with the pyrochlore structure, and, in particular, discuss ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, and extract its full set of Hamiltonian parameters from high-field inelastic neutron scattering experiments. We show that fluctuations in ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ are strong, and that the Hamiltonian may support a Coulombic “quantum spin liquid” ground state in low magnetic fields and host an unusual quantum critical point at larger fields. This appears consistent with puzzling features seen in prior experiments on ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. Thus, ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ is the first quantum spin liquid candidate for which the Hamiltonian is quantitatively known.

Here we establish the systematic existence of a $U(1)$ degeneracy of all symmetry-allowed Hamiltonians quadratic in the spins on the pyrochlore lattice, at the mean-field level. By extracting the Hamiltonian of ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ from inelastic neutron scattering measurements, we then show that the $U(1)$-degenerate states of ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ are its classical ground states, and unambiguously show that quantum fluctuations break the degeneracy in a way which is confirmed by experiment. The degree of symmetry protection of the classical $U(1)$ degeneracy in ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ is unprecedented in other materials. As a consequence, our observation of order by disorder is unusually definitive. We provide further verifiable consequences of this phenomenon, and several additional comparisons between theory and experiment.

Recent neutron scattering and specific heat studies on the pyrochlore Yb2Ti2O7 have revealed variations in its magnetic behavior below 265 mK. In the best samples, a sharp anomaly in the specific heat is observed at T=265~mK. Other samples, especially single crystals, have broad features in the specific heat which vary in sharpness and temperature depending on the sample, indicating that the magnetic ground state may be qualitatively different in such samples. We performed detailed comparisons of the chemical structure of a pulverized single crystal of Yb2Ti2O7, grown by the floating zone technique, to a sintered powder sample of Yb2Ti2O7. Rietveld refinements of neutron powder diffraction data on these samples reveal that the crushed single crystal is best described as a ``stuffed'' pyrochlore, Yb2(Ti2$-${}xYbx)O7$-${}x/2 with x~= 0.046(4), despite perfectly stoichiometric starting material. Substituting magnetic Yb3+ on the nonmagnetic Ti4+ sublattice would introduce random exchange bonds and local lattice deformations. These are expected to be the mechanism leading to the variation of the delicate magnetic ground state of Yb2Ti2O7. Determination of the cubic cell length a could be useful as a method for characterizing the stoichiometry of nonpulverized single crystals at room temperature.

Looking for a crystalline 2D spin ice: Spin ices—materials in which local magnetic spins respect “ice rules” similar to those in water ice—are typically three-dimensional. Two-dimensional (2D) ice rules can also be formulated and have been found to be satisfied in engineered nanomagnetic systems, usually referred to as artificial spin ices. Zhao et al. used neutron scattering and thermodynamic measurements to study a crystalline candidate for a 2D spin ice, the intermetallic compound HoAgGe. They found that at low temperatures, the local spins on the distorted kagome planes of this quasi-2D material respect 2D ice rules. Increasing the temperature led to a series of transitions consistent with theoretical expectations. Science , this issue p. 1218

Neutron scattering measurements show the ferromagnetic $XY$ pyrochlore ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ to display strong quasi-two-dimensional (2D) spin correlations at low temperature, which give way to long range order (LRO) under the application of modest magnetic fields. Rods of scattering along $⟨111⟩$ directions due to these 2D spin correlations imply a magnetic decomposition of the cubic pyrochlore system into decoupled kagome planes. A magnetic field of $\sim 0.5\mathrm{T}$ applied along the $[1\overline{1}0]$ direction induces a transition to a 3D LRO state characterized by long-lived, dispersive spin waves. Our measurements map out a complex low temperature-field phase diagram for this exotic pyrochlore magnet.

The pyrochlore material Yb${}_{2}$Ti${}_{2}$O${}_{7}$ displays unexpected quasi-two-dimensional (2D) magnetic correlations within a cubic lattice environment at low temperatures, before entering an exotic disordered ground state below $T=265$ mK. We report neutron scattering measurements of the thermal evolution of the 2D spin correlations in space and time. Short-range three-dimensional (3D) spin correlations develop below 400 mK, accompanied by a suppression in the quasielastic (QE) scattering below $$\sim${}$0.2 meV. These show a slowly fluctuating ground state with spins correlated over short distances within a kagome-triangular-kagome (KTK) stack along [111], which evolves to isolated kagome spin stars at higher temperatures. Furthermore, low-temperature specific heat results indicate a sample dependence to the putative transition temperature that is bounded by 265 mK, which we discuss in the context of recent mean field theoretical analysis.